Copyright © Cengage Learning. All rights reserved. 4 Polynomials Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. Section Dividing Polynomials by Monomials 4.7 Copyright © Cengage Learning. All rights reserved.

Objectives 1 Divide a monomial by a monomial. Divide a polynomial by a monomial. Solve a formula for a specified variable. 2 3

Divide a monomial by a monomial 1.

Divide a monomial by a monomial We have seen that dividing by a number is equivalent to multiplying by its reciprocal. For example, dividing the number 8 by 2 gives the same answer as multiplying 8 by and

Divide a monomial by a monomial In general, the following is true. Division Recall that to simplify a fraction, we write both its numerator and its denominator as the product of several factors and then divide out all common factors.

Divide a monomial by a monomial For example,

Divide a monomial by a monomial We can use the same method to simplify algebraic fractions that contain variables. We must assume, however, that no variable is 0. Factor: p2 = p  p, 6 = 2  3, and q3 = q  q  q. Divide out the common factors of 3, p, and q.

Divide a monomial by a monomial To divide monomials, we can either use the previous method or use the rules of exponents. Comment In all examples and exercises in this section, we will assume that no variables are 0 to avoid the possibility of division by 0. Recall that division by 0 is undefined.

Example Simplify. Solution: Using Fractions Using the Rules of Exponents

Example – Solution cont’d Using Fractions Using the Rules of Exponents

Divide a polynomial by a monomial 2.

Divide a polynomial by a monomial We saw that Since this is true, we also have This suggests that, to divide a polynomial by a monomial, we can divide each term of the polynomial in the numerator by the monomial in the denominator.

Example Simplify: Solution: Divide each term in the numerator by the monomial. Simplify each fraction.

Solve a formula for a specified variable 3.

Solve a formula for a specified variable The cross-sectional area of the trapezoidal drainage ditch shown in Figure 4-7 is given by the formula where B and b are its bases and h is its height. To solve the formula for b, we proceed as follows. Figure 4-7

Solve a formula for a specified variable A = h(B + b) 2(A) = 2  h(B + b) 2A = h(B + b) 2A = hB + hb 2A – hB = hB – hB + hb 2A – hB = hb Multiply both sides by 2 to clear fractions. Simplify. Use the distributive property to remove parentheses. Subtract hB from both sides. Combine like terms. Divide both sides by h.

Example Another student worked the previous problem in a different way and got a result of Is this also correct? Solution: To show that this result is correct, we must show that

Example – Solution cont’d We can do this by dividing 2A – hB by h. The results are the same. Divide each term in the numerator by the monomial. Simplify: